Diagrams.Solve.Polynomial:quartForm from diagrams-solve-0.1, C

Percentage Accurate: 97.6% → 97.6%

Time: 3.5s

Alternatives: 12

Speedup: 1.0×

• 32.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((x * y) + ((z * t) / 16.0d0)) - ((a * b) / 4.0d0)) + c
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

Python

def code(x, y, z, t, a, b, c):
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0)) - Float64(Float64(a * b) / 4.0)) + c)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]

Initial Program: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((x * y) + ((z * t) / 16.0d0)) - ((a * b) / 4.0d0)) + c
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

Python

def code(x, y, z, t, a, b, c):
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0)) - Float64(Float64(a * b) / 4.0)) + c)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]

Alternative 1: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((x * y) + ((z * t) / 16.0d0)) - ((a * b) / 4.0d0)) + c
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

Python

def code(x, y, z, t, a, b, c):
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0)) - Float64(Float64(a * b) / 4.0)) + c)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]

Derivation

1. Initial program 97.6%

   \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
2. Add Preprocessing

Alternative 2: 89.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.25 \cdot \left(b \cdot a\right)\\ t_2 := \frac{z \cdot t}{16}\\ t_3 := \mathsf{fma}\left(t \cdot z, 0.0625, c\right) - t\_1\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+70}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 15000000000:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* 0.25 (* b a)))
        (t_2 (/ (* z t) 16.0))
        (t_3 (- (fma (* t z) 0.0625 c) t_1)))
   (if (<= t_2 -1e+70)
     t_3
     (if (<= t_2 15000000000.0) (- (fma y x c) t_1) t_3))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = 0.25 * (b * a);
	double t_2 = (z * t) / 16.0;
	double t_3 = fma((t * z), 0.0625, c) - t_1;
	double tmp;
	if (t_2 <= -1e+70) {
		tmp = t_3;
	} else if (t_2 <= 15000000000.0) {
		tmp = fma(y, x, c) - t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(0.25 * Float64(b * a))
	t_2 = Float64(Float64(z * t) / 16.0)
	t_3 = Float64(fma(Float64(t * z), 0.0625, c) - t_1)
	tmp = 0.0
	if (t_2 <= -1e+70)
		tmp = t_3;
	elseif (t_2 <= 15000000000.0)
		tmp = Float64(fma(y, x, c) - t_1);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+70], t$95$3, If[LessEqual[t$95$2, 15000000000.0], N[(N[(y * x + c), $MachinePrecision] - t$95$1), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -1.00000000000000007e70 or 1.5e10 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 95.7%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(c + \frac{1}{16} \cdot \left(t \cdot z\right)\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(c + \frac{1}{16} \cdot \left(t \cdot z\right)\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(t \cdot z\right) \cdot \frac{1}{16} + c\right) - \frac{1}{4} \cdot \left(a \cdot b\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, c\right) - \frac{1}{4} \cdot \left(a \cdot b\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, c\right) - \frac{1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, c\right) - \frac{1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

      8. lower-*.f64 81.6

         \[\leadsto \mathsf{fma}\left(t \cdot z, 0.0625, c\right) - 0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

   4. Applied rewrites 81.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot z, 0.0625, c\right) - 0.25 \cdot \left(b \cdot a\right)} \]

   if -1.00000000000000007e70 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 1.5e10

   1. Initial program 98.9%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(c + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot y + c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      3. *-commutative N/A

         \[\leadsto \left(y \cdot x + c\right) - \frac{1}{4} \cdot \left(a \cdot b\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

      7. lower-*.f64 94.5

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

   4. Applied rewrites 94.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 89.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, 0.0625, -0.25 \cdot \left(b \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)))
   (if (<= t_1 -1e+70)
     (fma (* 0.0625 t) z (fma y x c))
     (if (<= t_1 5e+95)
       (- (fma y x c) (* 0.25 (* b a)))
       (fma (* t z) 0.0625 (* -0.25 (* b a)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double tmp;
	if (t_1 <= -1e+70) {
		tmp = fma((0.0625 * t), z, fma(y, x, c));
	} else if (t_1 <= 5e+95) {
		tmp = fma(y, x, c) - (0.25 * (b * a));
	} else {
		tmp = fma((t * z), 0.0625, (-0.25 * (b * a)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	tmp = 0.0
	if (t_1 <= -1e+70)
		tmp = fma(Float64(0.0625 * t), z, fma(y, x, c));
	elseif (t_1 <= 5e+95)
		tmp = Float64(fma(y, x, c) - Float64(0.25 * Float64(b * a)));
	else
		tmp = fma(Float64(t * z), 0.0625, Float64(-0.25 * Float64(b * a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+70], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+95], N[(N[(y * x + c), $MachinePrecision] - N[(0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t * z), $MachinePrecision] * 0.0625 + N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -1.00000000000000007e70

   1. Initial program 95.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

   4. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)}{a} - 0.25 \cdot b\right) \cdot a} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) + \color{blue}{c} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + x \cdot y\right) + c \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + y \cdot x\right) + c \]

      4. associate-+r+ N/A

         \[\leadsto \left(\frac{1}{16} \cdot t\right) \cdot z + \color{blue}{\left(y \cdot x + c\right)} \]

      5. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, \color{blue}{z}, y \cdot x + c\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, y \cdot x + c\right) \]

      7. lift-fma.f64 82.9

         \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right) \]

   7. Applied rewrites 82.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)} \]

   if -1.00000000000000007e70 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 5.00000000000000025e95

   1. Initial program 98.9%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(c + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot y + c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      3. *-commutative N/A

         \[\leadsto \left(y \cdot x + c\right) - \frac{1}{4} \cdot \left(a \cdot b\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

      7. lower-*.f64 93.1

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

   4. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)} \]

   if 5.00000000000000025e95 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 95.4%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(c + \frac{1}{16} \cdot \left(t \cdot z\right)\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(c + \frac{1}{16} \cdot \left(t \cdot z\right)\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(t \cdot z\right) \cdot \frac{1}{16} + c\right) - \frac{1}{4} \cdot \left(a \cdot b\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, c\right) - \frac{1}{4} \cdot \left(a \cdot b\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, c\right) - \frac{1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, c\right) - \frac{1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

      8. lower-*.f64 82.8

         \[\leadsto \mathsf{fma}\left(t \cdot z, 0.0625, c\right) - 0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

   4. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot z, 0.0625, c\right) - 0.25 \cdot \left(b \cdot a\right)} \]

   5. Taylor expanded in c around 0

      \[\leadsto \frac{1}{16} \cdot \left(t \cdot z\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{1}{16} \cdot \left(t \cdot z\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \color{blue}{\left(a \cdot b\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(t \cdot z\right) \cdot \frac{1}{16} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(\color{blue}{a} \cdot b\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(t \cdot z\right) \cdot \frac{1}{16} + \frac{-1}{4} \cdot \left(a \cdot b\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, \frac{-1}{4} \cdot \left(b \cdot a\right)\right) \]

      8. lift-*.f64 75.1

         \[\leadsto \mathsf{fma}\left(t \cdot z, 0.0625, -0.25 \cdot \left(b \cdot a\right)\right) \]

   7. Applied rewrites 75.1%

      \[\leadsto \mathsf{fma}\left(t \cdot z, \color{blue}{0.0625}, -0.25 \cdot \left(b \cdot a\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 88.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ t_2 := \mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+70}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)) (t_2 (fma (* 0.0625 t) z (fma y x c))))
   (if (<= t_1 -1e+70)
     t_2
     (if (<= t_1 1e+48) (- (fma y x c) (* 0.25 (* b a))) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double t_2 = fma((0.0625 * t), z, fma(y, x, c));
	double tmp;
	if (t_1 <= -1e+70) {
		tmp = t_2;
	} else if (t_1 <= 1e+48) {
		tmp = fma(y, x, c) - (0.25 * (b * a));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	t_2 = fma(Float64(0.0625 * t), z, fma(y, x, c))
	tmp = 0.0
	if (t_1 <= -1e+70)
		tmp = t_2;
	elseif (t_1 <= 1e+48)
		tmp = Float64(fma(y, x, c) - Float64(0.25 * Float64(b * a)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x + c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+70], t$95$2, If[LessEqual[t$95$1, 1e+48], N[(N[(y * x + c), $MachinePrecision] - N[(0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -1.00000000000000007e70 or 1.00000000000000004e48 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 95.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

   4. Applied rewrites 80.6%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)}{a} - 0.25 \cdot b\right) \cdot a} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) + \color{blue}{c} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + x \cdot y\right) + c \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + y \cdot x\right) + c \]

      4. associate-+r+ N/A

         \[\leadsto \left(\frac{1}{16} \cdot t\right) \cdot z + \color{blue}{\left(y \cdot x + c\right)} \]

      5. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, \color{blue}{z}, y \cdot x + c\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, y \cdot x + c\right) \]

      7. lift-fma.f64 82.9

         \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right) \]

   7. Applied rewrites 82.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)} \]

   if -1.00000000000000007e70 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 1.00000000000000004e48

   1. Initial program 98.9%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(c + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot y + c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      3. *-commutative N/A

         \[\leadsto \left(y \cdot x + c\right) - \frac{1}{4} \cdot \left(a \cdot b\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

      7. lower-*.f64 93.8

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

   4. Applied rewrites 93.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 87.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ t_2 := \mathsf{fma}\left(-0.25, b \cdot a, y \cdot x\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+129}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)) (t_2 (fma -0.25 (* b a) (* y x))))
   (if (<= t_1 -5e+129)
     t_2
     (if (<= t_1 1e+140) (fma (* 0.0625 t) z (fma y x c)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double t_2 = fma(-0.25, (b * a), (y * x));
	double tmp;
	if (t_1 <= -5e+129) {
		tmp = t_2;
	} else if (t_1 <= 1e+140) {
		tmp = fma((0.0625 * t), z, fma(y, x, c));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	t_2 = fma(-0.25, Float64(b * a), Float64(y * x))
	tmp = 0.0
	if (t_1 <= -5e+129)
		tmp = t_2;
	elseif (t_1 <= 1e+140)
		tmp = fma(Float64(0.0625 * t), z, fma(y, x, c));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, Block[{t$95$2 = N[(-0.25 * N[(b * a), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+129], t$95$2, If[LessEqual[t$95$1, 1e+140], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x + c), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -5.0000000000000003e129 or 1.00000000000000006e140 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 93.9%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(c + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot y + c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      3. *-commutative N/A

         \[\leadsto \left(y \cdot x + c\right) - \frac{1}{4} \cdot \left(a \cdot b\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

      7. lower-*.f64 85.5

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

   4. Applied rewrites 85.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)} \]

   5. Taylor expanded in c around 0

      \[\leadsto x \cdot y - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto x \cdot y + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \color{blue}{\left(a \cdot b\right)} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot y + \frac{-1}{4} \cdot \left(a \cdot b\right) \]

      3. +-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \left(a \cdot b\right) + x \cdot \color{blue}{y} \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, a \cdot \color{blue}{b}, x \cdot y\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, b \cdot a, x \cdot y\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, b \cdot a, x \cdot y\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, b \cdot a, y \cdot x\right) \]

      8. lift-*.f64 79.4

         \[\leadsto \mathsf{fma}\left(-0.25, b \cdot a, y \cdot x\right) \]

   7. Applied rewrites 79.4%

      \[\leadsto \mathsf{fma}\left(-0.25, \color{blue}{b \cdot a}, y \cdot x\right) \]

   if -5.0000000000000003e129 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 1.00000000000000006e140

   1. Initial program 99.2%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

   4. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)}{a} - 0.25 \cdot b\right) \cdot a} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) + \color{blue}{c} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + x \cdot y\right) + c \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + y \cdot x\right) + c \]

      4. associate-+r+ N/A

         \[\leadsto \left(\frac{1}{16} \cdot t\right) \cdot z + \color{blue}{\left(y \cdot x + c\right)} \]

      5. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, \color{blue}{z}, y \cdot x + c\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, y \cdot x + c\right) \]

      7. lift-fma.f64 90.8

         \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right) \]

   7. Applied rewrites 90.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 77.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(0.0625 \cdot t, z, y \cdot x\right)\\ t_2 := x \cdot y + \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+158}:\\ \;\;\;\;c - 0.25 \cdot \left(b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* 0.0625 t) z (* y x))) (t_2 (+ (* x y) (/ (* z t) 16.0))))
   (if (<= t_2 -1e+59) t_1 (if (<= t_2 5e+158) (- c (* 0.25 (* b a))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma((0.0625 * t), z, (y * x));
	double t_2 = (x * y) + ((z * t) / 16.0);
	double tmp;
	if (t_2 <= -1e+59) {
		tmp = t_1;
	} else if (t_2 <= 5e+158) {
		tmp = c - (0.25 * (b * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(0.0625 * t), z, Float64(y * x))
	t_2 = Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0))
	tmp = 0.0
	if (t_2 <= -1e+59)
		tmp = t_1;
	elseif (t_2 <= 5e+158)
		tmp = Float64(c - Float64(0.25 * Float64(b * a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+59], t$95$1, If[LessEqual[t$95$2, 5e+158], N[(c - N[(0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -9.99999999999999972e58 or 4.9999999999999996e158 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64)))

   1. Initial program 95.7%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{a} + \left(\frac{c}{a} + \frac{x \cdot y}{a}\right)\right) - \frac{1}{4} \cdot b\right) \cdot \color{blue}{a} \]

   4. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)}{a} - 0.25 \cdot b\right) \cdot a} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) + \color{blue}{c} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + x \cdot y\right) + c \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + y \cdot x\right) + c \]

      4. associate-+r+ N/A

         \[\leadsto \left(\frac{1}{16} \cdot t\right) \cdot z + \color{blue}{\left(y \cdot x + c\right)} \]

      5. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, \color{blue}{z}, y \cdot x + c\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, y \cdot x + c\right) \]

      7. lift-fma.f64 84.8

         \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right) \]

   7. Applied rewrites 84.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, x \cdot y\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, y \cdot x\right) \]

      2. lift-*.f64 76.3

         \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, y \cdot x\right) \]

   10. Applied rewrites 76.3%

       \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, y \cdot x\right) \]

   if -9.99999999999999972e58 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 4.9999999999999996e158

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(c + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot y + c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      3. *-commutative N/A

         \[\leadsto \left(y \cdot x + c\right) - \frac{1}{4} \cdot \left(a \cdot b\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

      7. lower-*.f64 89.8

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

   4. Applied rewrites 89.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto c - \color{blue}{\frac{1}{4}} \cdot \left(b \cdot a\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 78.9%

      \[\leadsto c - \color{blue}{0.25} \cdot \left(b \cdot a\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 63.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ t_2 := \left(t \cdot z\right) \cdot 0.0625\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+63}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-114}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, y \cdot x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+50}:\\ \;\;\;\;c - 0.25 \cdot \left(b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)) (t_2 (* (* t z) 0.0625)))
   (if (<= t_1 -5e+63)
     t_2
     (if (<= t_1 -1e-114)
       (fma -0.25 (* b a) (* y x))
       (if (<= t_1 2e+50) (- c (* 0.25 (* b a))) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double t_2 = (t * z) * 0.0625;
	double tmp;
	if (t_1 <= -5e+63) {
		tmp = t_2;
	} else if (t_1 <= -1e-114) {
		tmp = fma(-0.25, (b * a), (y * x));
	} else if (t_1 <= 2e+50) {
		tmp = c - (0.25 * (b * a));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	t_2 = Float64(Float64(t * z) * 0.0625)
	tmp = 0.0
	if (t_1 <= -5e+63)
		tmp = t_2;
	elseif (t_1 <= -1e-114)
		tmp = fma(-0.25, Float64(b * a), Float64(y * x));
	elseif (t_1 <= 2e+50)
		tmp = Float64(c - Float64(0.25 * Float64(b * a)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+63], t$95$2, If[LessEqual[t$95$1, -1e-114], N[(-0.25 * N[(b * a), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+50], N[(c - N[(0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -5.00000000000000011e63 or 2.0000000000000002e50 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 95.5%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t \cdot z\right) \cdot \color{blue}{\frac{1}{16}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(t \cdot z\right) \cdot \color{blue}{\frac{1}{16}} \]

      3. lower-*.f64 57.7

         \[\leadsto \left(t \cdot z\right) \cdot 0.0625 \]

   4. Applied rewrites 57.7%

      \[\leadsto \color{blue}{\left(t \cdot z\right) \cdot 0.0625} \]

   if -5.00000000000000011e63 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -1.0000000000000001e-114

   1. Initial program 99.1%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(c + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot y + c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      3. *-commutative N/A

         \[\leadsto \left(y \cdot x + c\right) - \frac{1}{4} \cdot \left(a \cdot b\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

      7. lower-*.f64 87.3

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

   4. Applied rewrites 87.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)} \]

   5. Taylor expanded in c around 0

      \[\leadsto x \cdot y - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto x \cdot y + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \color{blue}{\left(a \cdot b\right)} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot y + \frac{-1}{4} \cdot \left(a \cdot b\right) \]

      3. +-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \left(a \cdot b\right) + x \cdot \color{blue}{y} \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, a \cdot \color{blue}{b}, x \cdot y\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, b \cdot a, x \cdot y\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, b \cdot a, x \cdot y\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, b \cdot a, y \cdot x\right) \]

      8. lift-*.f64 59.6

         \[\leadsto \mathsf{fma}\left(-0.25, b \cdot a, y \cdot x\right) \]

   7. Applied rewrites 59.6%

      \[\leadsto \mathsf{fma}\left(-0.25, \color{blue}{b \cdot a}, y \cdot x\right) \]

   if -1.0000000000000001e-114 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 2.0000000000000002e50

   1. Initial program 98.9%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(c + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot y + c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      3. *-commutative N/A

         \[\leadsto \left(y \cdot x + c\right) - \frac{1}{4} \cdot \left(a \cdot b\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

      7. lower-*.f64 96.0

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

   4. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto c - \color{blue}{\frac{1}{4}} \cdot \left(b \cdot a\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 64.8%

      \[\leadsto c - \color{blue}{0.25} \cdot \left(b \cdot a\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 61.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ t_2 := \left(t \cdot z\right) \cdot 0.0625\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+70}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+50}:\\ \;\;\;\;c - 0.25 \cdot \left(b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)) (t_2 (* (* t z) 0.0625)))
   (if (<= t_1 -1e+70)
     t_2
     (if (<= t_1 -5e-35)
       (fma y x c)
       (if (<= t_1 2e+50) (- c (* 0.25 (* b a))) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double t_2 = (t * z) * 0.0625;
	double tmp;
	if (t_1 <= -1e+70) {
		tmp = t_2;
	} else if (t_1 <= -5e-35) {
		tmp = fma(y, x, c);
	} else if (t_1 <= 2e+50) {
		tmp = c - (0.25 * (b * a));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	t_2 = Float64(Float64(t * z) * 0.0625)
	tmp = 0.0
	if (t_1 <= -1e+70)
		tmp = t_2;
	elseif (t_1 <= -5e-35)
		tmp = fma(y, x, c);
	elseif (t_1 <= 2e+50)
		tmp = Float64(c - Float64(0.25 * Float64(b * a)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+70], t$95$2, If[LessEqual[t$95$1, -5e-35], N[(y * x + c), $MachinePrecision], If[LessEqual[t$95$1, 2e+50], N[(c - N[(0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -1.00000000000000007e70 or 2.0000000000000002e50 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 95.5%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t \cdot z\right) \cdot \color{blue}{\frac{1}{16}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(t \cdot z\right) \cdot \color{blue}{\frac{1}{16}} \]

      3. lower-*.f64 58.1

         \[\leadsto \left(t \cdot z\right) \cdot 0.0625 \]

   4. Applied rewrites 58.1%

      \[\leadsto \color{blue}{\left(t \cdot z\right) \cdot 0.0625} \]

   if -1.00000000000000007e70 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -4.99999999999999964e-35

   1. Initial program 98.8%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(c + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot y + c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      3. *-commutative N/A

         \[\leadsto \left(y \cdot x + c\right) - \frac{1}{4} \cdot \left(a \cdot b\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

      7. lower-*.f64 82.9

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

   4. Applied rewrites 82.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto c + \color{blue}{x \cdot y} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto c + y \cdot x \]

      2. +-commutative N/A

         \[\leadsto y \cdot x + c \]

      3. lift-fma.f64 54.8

         \[\leadsto \mathsf{fma}\left(y, x, c\right) \]

   7. Applied rewrites 54.8%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{x}, c\right) \]

   if -4.99999999999999964e-35 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 2.0000000000000002e50

   1. Initial program 98.9%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(c + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot y + c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      3. *-commutative N/A

         \[\leadsto \left(y \cdot x + c\right) - \frac{1}{4} \cdot \left(a \cdot b\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

      7. lower-*.f64 95.6

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

   4. Applied rewrites 95.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto c - \color{blue}{\frac{1}{4}} \cdot \left(b \cdot a\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 64.8%

      \[\leadsto c - \color{blue}{0.25} \cdot \left(b \cdot a\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 61.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ t_2 := -0.25 \cdot \left(b \cdot a\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+129}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)) (t_2 (* -0.25 (* b a))))
   (if (<= t_1 -5e+129) t_2 (if (<= t_1 5e+87) (fma y x c) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double t_2 = -0.25 * (b * a);
	double tmp;
	if (t_1 <= -5e+129) {
		tmp = t_2;
	} else if (t_1 <= 5e+87) {
		tmp = fma(y, x, c);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	t_2 = Float64(-0.25 * Float64(b * a))
	tmp = 0.0
	if (t_1 <= -5e+129)
		tmp = t_2;
	elseif (t_1 <= 5e+87)
		tmp = fma(y, x, c);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, Block[{t$95$2 = N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+129], t$95$2, If[LessEqual[t$95$1, 5e+87], N[(y * x + c), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -5.0000000000000003e129 or 4.9999999999999998e87 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 94.4%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

      3. lower-*.f64 66.5

         \[\leadsto -0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

   4. Applied rewrites 66.5%

      \[\leadsto \color{blue}{-0.25 \cdot \left(b \cdot a\right)} \]

   if -5.0000000000000003e129 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 4.9999999999999998e87

   1. Initial program 99.2%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(c + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot y + c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      3. *-commutative N/A

         \[\leadsto \left(y \cdot x + c\right) - \frac{1}{4} \cdot \left(a \cdot b\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

      7. lower-*.f64 69.3

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

   4. Applied rewrites 69.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto c + \color{blue}{x \cdot y} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto c + y \cdot x \]

      2. +-commutative N/A

         \[\leadsto y \cdot x + c \]

      3. lift-fma.f64 61.8

         \[\leadsto \mathsf{fma}\left(y, x, c\right) \]

   7. Applied rewrites 61.8%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{x}, c\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 48.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, x, c\right) \end{array} \]

FPCore

(FPCore (x y z t a b c) :precision binary64 (fma y x c))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return fma(y, x, c);
}

Julia

function code(x, y, z, t, a, b, c)
	return fma(y, x, c)
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(y * x + c), $MachinePrecision]

Derivation

1. Initial program 97.6%

   \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

2. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
3. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(c + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]

   2. +-commutative N/A

      \[\leadsto \left(x \cdot y + c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

   3. *-commutative N/A

      \[\leadsto \left(y \cdot x + c\right) - \frac{1}{4} \cdot \left(a \cdot b\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(y, x, c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

   7. lower-*.f64 74.5

      \[\leadsto \mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

4. Applied rewrites 74.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)} \]

5. Taylor expanded in a around 0

   \[\leadsto c + \color{blue}{x \cdot y} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto c + y \cdot x \]

   2. +-commutative N/A

      \[\leadsto y \cdot x + c \]

   3. lift-fma.f64 48.7

      \[\leadsto \mathsf{fma}\left(y, x, c\right) \]

7. Applied rewrites 48.7%

   \[\leadsto \mathsf{fma}\left(y, \color{blue}{x}, c\right) \]
8. Add Preprocessing

Alternative 11: 40.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+99}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \cdot y \leq 10^{-45}:\\ \;\;\;\;c\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -5e+99) (* y x) (if (<= (* x y) 1e-45) c (* y x))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -5e+99) {
		tmp = y * x;
	} else if ((x * y) <= 1e-45) {
		tmp = c;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if ((x * y) <= (-5d+99)) then
        tmp = y * x
    else if ((x * y) <= 1d-45) then
        tmp = c
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -5e+99) {
		tmp = y * x;
	} else if ((x * y) <= 1e-45) {
		tmp = c;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if (x * y) <= -5e+99:
		tmp = y * x
	elif (x * y) <= 1e-45:
		tmp = c
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x * y) <= -5e+99)
		tmp = Float64(y * x);
	elseif (Float64(x * y) <= 1e-45)
		tmp = c;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if ((x * y) <= -5e+99)
		tmp = y * x;
	elseif ((x * y) <= 1e-45)
		tmp = c;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x * y), $MachinePrecision], -5e+99], N[(y * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e-45], c, N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -5.00000000000000008e99 or 9.99999999999999984e-46 < (*.f64 x y)

   1. Initial program 96.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot y} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{x} \]

      2. lower-*.f64 53.2

         \[\leadsto y \cdot \color{blue}{x} \]

   4. Applied rewrites 53.2%

      \[\leadsto \color{blue}{y \cdot x} \]

   if -5.00000000000000008e99 < (*.f64 x y) < 9.99999999999999984e-46

   1. Initial program 98.9%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c} \]
   3. Step-by-step derivation

   4. Applied rewrites 29.8%

      \[\leadsto \color{blue}{c} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 22.7% accurate, 24.7× speedup

Math

\[\begin{array}{l} \\ c \end{array} \]

FPCore

(FPCore (x y z t a b c) :precision binary64 c)

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return c;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return c;
}

Python

def code(x, y, z, t, a, b, c):
	return c

Julia

function code(x, y, z, t, a, b, c)
	return c
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = c;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := c

Derivation

1. Initial program 97.6%

   \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

2. Taylor expanded in c around inf

   \[\leadsto \color{blue}{c} \]
3. Step-by-step derivation

4. Applied rewrites 22.7%

   \[\leadsto \color{blue}{c} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025117 
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, C"
  :precision binary64
  (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))